Deformations and homotopy of Rota-Baxter operators and $\mathcal{O}$-operators on Lie algebras

TL;DR

This paper explores the deformation and homotopy theories of Rota-Baxter and $ ext{O}$-operators on Lie algebras using differential graded Lie algebra methods, extending their connection to pre-Lie algebras.

Contribution

It introduces a differential graded Lie algebra framework for studying deformations and homotopies of Rota-Baxter and $ ext{O}$-operators, expanding their algebraic relationships.

Findings

Deformation theory of Rota-Baxter and $ ext{O}$-operators is developed.

Homotopy theories for these operators are established.

Connections to pre-Lie algebras are extended to deformation and homotopy levels.

Abstract

This article gives a brief introduction to some recent work on deformation and homotopy theories of Rota-Baxter operators and more generally $\mathcal{O}$-operators on Lie algebras, by means of the differential graded Lie algebra approach. It is further shown that these theories lift the existing connection between $\mathcal{O}$-operators and pre-Lie algebras to the levels of deformations and homotopy.